Sagot :
Given:
The graph passes through the given points (4,-2) and (2,0).
To find:
The quadratic function [tex]y=a(x-h)^2[/tex].
Solution:
We have, quadratic function
[tex]y=a(x-h)^2[/tex] ...(i)
The graph passes through the given points (4,-2) and (2,0). It means the equation must be true for these points.
Putting x=4 and y=-2 in (i), we get
[tex]-2=a(4-h)^2[/tex] ...(ii)
Putting x=2 and y=0 in (i), we get
[tex]0=a(2-h)^2[/tex] ...(iii)
Divide (iii) by (ii).
[tex]\dfrac{0}{-2}=\dfrac{a(2-h)^2}{a(4-h)^2}[/tex]
[tex]0=\dfrac{(2-h)^2}{(4-h)^2}[/tex]
[tex]0=(2-h)^2[/tex]
[tex]0=2-h[/tex]
[tex]h=2[/tex]
Putting h=2 in (ii), we get
[tex]-2=a(4-2)^2[/tex]
[tex]-2=a(2)^2[/tex]
[tex]-2=a(4)[/tex]
Divide both sides by 4.
[tex]\dfrac{-2}{4}=a[/tex]
[tex]\dfrac{-1}{2}=a[/tex]
Putting [tex]a=-\dfrac{1}{2}[/tex] and h=2 in (i), we get
[tex]y=-\dfrac{1}{2}(x-2)^2[/tex]
Therefore, the required quadratic function is [tex]y=-\dfrac{1}{2}(x-2)^2[/tex].